Were the Brotherhood of Steel and Enclave part of the USA government?. WHy do they fight each other?

BOS origins was from soldiers disgusted with the experiments what was going on with Mariposa Military base (where super mutants were being made) and turned on the people there before leaving shortly after the bombs fell and set up base at Lost Hills Bunker eventually forming the Brotherhood of Steel

Enclave was more a small off shoot of the government (which was quite a small secretive and selective few- Even if FO3 portrayed them as an unlimited army)) who fled and hid off a western coast oil rig and probably a few bunkers around the country and after the bombs fell and was safe to leave with the goal of recreating the United States by killing off all mutation though various large scale poisoning of the water supply with chemicals designed to kill off mutated ells etc... so from Super Mutants to humans brought up in the wastes

So BOS and Enclave really haven't got much connection and if I remember rightly the BOS for the most part have forgotten about there origins and even then Enclave would never accept them purely because they genetically inferior

Here are some videos you can watch to explain this. DO NOT WORRY!!! There is no virus.

https://www.youtube.com/watch?v=seKGaWVWKyc&index=4&list=PL7pGJQV-jlzD17YNNbt103xp0PkkUCoPU

https://www.youtube.com/watch?v=sGLPBEkW0Ps&list=PL7pGJQV-jlzD17YNNbt103xp0PkkUCoPU&index=5

https://www.youtube.com/watch?v=bnjXyGcU59k&index=6&list=PL7pGJQV-jlzD17YNNbt103xp0PkkUCoPU

Watch these and all will be explained.